On the Farrell Cohomology of Mapping Class Groups
H. H. Glover, G. Mislin and Y. Xia



Introduction.

For $\Gamma$ any group of finite virtual cohomological dimension and a prime 
$p$ we say that $\Gamma$ is $p$-periodic, if there exists a positive integer 
$k$ such that the Farrell cohomology groups $\hat H^i(\Gamma;M)$ and 
$\hat H^{i+k}(\Gamma;M)$ have naturally isomorphic $p$-primary components 
for all $i \subset \mathbb Z$ and $\mathbb Z\Gamma$-modules $M$. The 
$p$-period of $\Gamma$ is defined as the least value of $k$ (cf. [B]). For
instance, if $\Gamma$ is $p$-torsion free, then $\Gamma$ is $p$-periodic of 
period one.

The mapping class group, $\Gamma_g$, is defined to be the group of path 
components of the group of orientation preserving homeomorphisms of the 
oriented closed surface $S_g$ of genus $g$. For instance, 
$\Gamma_1 = SL(2,\mathbb Z)$ and the cohomology is well known and easy to 
compute in this case. By writing $SL(2,\mathbb Z)$ as an amalgamated product 
of $\mathbb Z/4$ and $\mathbb Z/6$ over $\mathbb Z/2$, one finds
\[ \hat H^*(\Gamma_1;\mathbb Z) \cong (\mathbb Z/12)[x,x^{-1}] \]
with $x$ of degree two. Thus $\Gamma_1$ is $2$- and $3$-periodic, with periods 
equal to two.

It is well known that $\Gamma_g$ is of finite virtual cohomological dimension 
and, if $g > 1$, $vcd(\Gamma_g) = 4g - 5$ (cf. [H]). In the sequel we will 
always assume that $g > 1$. Recall from [B] that a group of finite vcd is 
$p$-periodic if and only if it does not contain a subgroup isomorphic to 
$\mathbb Z/p \times \mathbb Z/p$. Because for $g > 1$ the mapping class group 
$\Gamma_g$ contains always a subgroup isomorphic to $\mathbb Z/2 \times 
\mathbb Z/2$, $\Gamma_g$ is never $2$-periodic. However, for an odd prime $p$,
$\Gamma_g$ is $p$-periodic for almost all values of $g$. This corresponds to 
the intuitively obvious fact that it is hard to find two ``different''
homeomorphisms of order $p$ on $S_g$, which commute with each other. The 
third author determined in [X1] all the genera $g$ for which $\Gamma_g$ is
$p$-periodic. In particular, $\Gamma_g$ is $3$-periodic if and only if 
$g \not\equiv 1$ mod($3$). For an odd prime $p$ and genus $g \not\equiv 1$ 
mod($p$), $\Gamma_g$ is always $p$-periodic. Moreover, there are only finitely 
many ``exceptional values'' of $g$ with $g \equiv 1$ mod($p$) for which 
$\Gamma_g$ is $p$-periodic.

Recall that for a finite $p$-periodic group $G$ and $p$ an odd prime, the 
Sylow $p$-subgroup $G_p$ of $G$ is cyclic and, if $G_p \ne 1$, the $p$-period 
of $G$ equals $2|N(G_p):C(G_p)|, where $N(G_p)$ (respectively $C(G_p)$) 
denotes the normalizer (respectively centralizer) of $G_p$ in $G$; in 
particular, the $p$-period of $G$ divides $2(p -1)$. Unlike the case of 
finite groups , the $p$-period of a $p$-periodic {\it infinite\/} group, 
$p$ a fixed  prime, may be arbitrarily large. A simple example
is given by the group $\mathbb Z/p^n \rtimes \mathbb Z = \langle a, b | 
a^{p^n} = 1, bab^{-1} = a^{p+1} \rangle$. For an odd prime $p$, the
$p$-period of $\mathbb Z/p^n \rtimes \mathbb Z$ equals $2p^{n-1}$.

In this paper, however, we will show the surprising result that for a 
$p$-periodic mapping class group $\Gamma_g$, the $p$-period is bounded by 
$2(p -1)$. The precise theorem reads as follows.

THEOREM 1.  Let $p$ be an odd prime and assume that $\Gamma_g$ is 
$p$-periodic. Then the $p$-period of $\Gamma_g$ is given by
\[ lcm\{2[N(\pi):C(\pi)] | \pi \in S\} \]
where $\pi$ ranges over $S$ , a set of representatives of conjugacy classes 
of subgroups of order $p$ of $\Gamma_g$, and $N(\pi)$ (respectively $C(\pi)$) 
denotes the normalizer (respectively centralizer) of $\pi$ in $\Gamma_g$. 
In particular, the $p$-period of $\Gamma_g$ divides $2(p -1)$.

We use the convention that $lcm\{2[N(\pi):C(\pi)] | \pi \in S\} = 1$ in case 
$S$ is empty (the $p$-period of $\Gamma_g$ equals one in that case).